| VE.Jk.Tukey.Corr.Hajek {samplingVarEst} | R Documentation |
The Tukey (1958) jackknife variance estimator for the estimator of a correlation coefficient using the Hajek point estimator
Description
Computes the Quenouille(1956); Tukey (1958) jackknife variance estimator for the estimator of a correlation coefficient of two variables using the Hajek (1971) point estimator.
Usage
VE.Jk.Tukey.Corr.Hajek(VecY.s, VecX.s, VecPk.s, N, FPC= TRUE)Arguments
VecY.s |
vector of the variable of interest Y; its length is equal to |
VecX.s |
vector of the variable of interest X; its length is equal to |
VecPk.s |
vector of the first-order inclusion probabilities; its length is equal to |
N |
the population size. It must be an integer or a double-precision scalar with zero-valued fractional part. This information is utilised for the finite population correction only, see |
FPC |
logical value. If an ad hoc finite population correction |
Details
For the population correlation coefficient of two variables y and x:
C = \frac{\sum_{k\in U} (y_k - \bar{y})(x_k - \bar{x})}{\sqrt{\sum_{k\in U} (y_k - \bar{y})^2}\sqrt{\sum_{k\in U} (x_k - \bar{x})^2}}
the point estimator of C, assuming that N is unknown (see Sarndal et al., 1992, Sec. 5.9), is:
\hat{C}_{Hajek} = \frac{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})(x_k - \hat{\bar{x}}_{Hajek})}{\sqrt{\sum_{k\in s} w_k (y_k - \hat{\bar{y}}_{Hajek})^2}\sqrt{\sum_{k\in s} w_k (x_k - \hat{\bar{x}}_{Hajek})^2}}
where \hat{\bar{y}}_{Hajek} is the Hajek (1971) point estimator of the population mean \bar{y} = N^{-1} \sum_{k\in U} y_k,
\hat{\bar{y}}_{Hajek} = \frac{\sum_{k\in s} w_k y_k}{\sum_{k\in s} w_k}
and w_k=1/\pi_k with \pi_k denoting the inclusion probability of the k-th element in the sample s. The variance of \hat{C}_{Hajek} can be estimated by the Quenouille(1956); Tukey (1958) jackknife variance estimator (implemented by the current function):
\hat{V}(\hat{C}_{Hajek}) = \left(1-\frac{n}{N}\right)\frac{n-1}{n}\sum_{k\in s} \left( \hat{C}_{Hajek(k)}-\hat{C}_{Hajek} \right)^2
where \hat{C}_{Hajek(k)} has the same functional form as \hat{C}_{Hajek} but omitting the k-th element from the sample s.
Note that we are implementing the Tukey (1958) jackknife variance estimator using the ‘ad hoc’ finite population correction 1-n/N (see Shao and Tu, 1995; Wolter, 2007). If FPC=FALSE then the term 1-n/N is ommited from the above formula.
Value
The function returns a value for the estimated variance.
Author(s)
Emilio Lopez Escobar.
References
Hajek, J. (1971) Comment on An essay on the logical foundations of survey sampling by Basu, D. in Foundations of Statistical Inference (Godambe, V.P. and Sprott, D.A. eds.), p. 236. Holt, Rinehart and Winston.
Quenouille, M. H. (1956) Notes on bias in estimation. Biometrika, 43, 353–360.
Sarndal, C.-E. and Swensson, B. and Wretman, J. (1992) Model Assisted Survey Sampling. Springer-Verlag, Inc.
Shao, J. and Tu, D. (1995) The Jackknife and Bootstrap. Springer-Verlag, Inc.
Tukey, J. W. (1958) Bias and confidence in not-quite large samples (abstract). The Annals of Mathematical Statistics, 29, 2, p. 614.
Wolter, K. M. (2007) Introduction to Variance Estimation. 2nd Ed. Springer, Inc.
See Also
VE.Jk.CBS.HT.Corr.HajekVE.Jk.CBS.SYG.Corr.HajekVE.Jk.B.Corr.HajekVE.Jk.EB.SW2.Corr.Hajek
Examples
data(oaxaca) #Loads the Oaxaca municipalities dataset
pik.U <- Pk.PropNorm.U(373, oaxaca$HOMES00) #Reconstructs the 1st order incl. probs.
s <- oaxaca$sHOMES00 #Defines the sample to be used
N <- dim(oaxaca)[1] #Defines the population size
y1 <- oaxaca$POP10 #Defines the variable of interest y1
y2 <- oaxaca$POPMAL10 #Defines the variable of interest y2
x <- oaxaca$HOMES10 #Defines the variable of interest x
#Computes the var. est. of the corr. coeff. point estimator using y1
VE.Jk.Tukey.Corr.Hajek(y1[s==1], x[s==1], pik.U[s==1], N)
#Computes the var. est. of the corr. coeff. point estimator using y2
VE.Jk.Tukey.Corr.Hajek(y2[s==1], x[s==1], pik.U[s==1], N, FPC= FALSE)